AIR FORCE INSTITUTE OF TECHNOLOGY

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1 OPTIMAL COVERAGE OF THEATER TARGETS WITH SMALL SATELLITE CONSTELLATIONS THESIS Axel Rendon, 2d Lt, USAF AFIT/GSS/ENY/06-M12 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

2 The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government.

3 AFIT/GSS/ENY/06-M12 OPTIMAL COVERAGE OF THEATER TARGETS WITH SMALL SATELLITE CONSTELLATIONS THESIS Presented to the Faculty Department of Aeronautics and Astronautics Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command In Partial Fulfillment of the Requirements for the Degree of Master of Science (Space Systems) Rendon, Axel, BS Second Lieutenant, USAF March 2006 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

4 AFIT/GSS/ENY/06-M12 OPTIMAL COVERAGE OF THEATER TARGETS WITH SMALL SATELLITE CONSTELLATIONS Rendon, Axel, BS Second Lieutenant, USAF Approved: /APPROVED/ 16 Mar 06 Nathan A. Titus (Chairman) date /APPROVED/ 16 Mar 06 Kerry D. Hicks (Member) date /APPROVED/ 16 Mar 06 William E. Wiesel (Member) date

5 Abstract The daylight passes of a low-earth orbit satellite over a targeted latitude and longitude are optimized by varying the inclination and eccentricity of an orbit at different altitudes. This investigation extends the work by Emery et al, in which the optimal Right Ascension of the Ascending Node was determined for a circular, matched inclination orbit. The optimal values were determined by a numerical research method based on Emery et al. s Matlab program. Results indicate that small increases in inclination raise the number of daylight passes up to 33%. These optimal inclinations depend on the satellite semi-major axis. Eccentricity increases also improve daylight pass numbers, but at a cost of increased range to the target. iv

6 Acknowledgments I would like to thank my family for their continuous wisdom and support that has guided me to reach this point. I would like to express my gratitude to my faculty advisor Lt Col. Nathan Titus because without his help and guidance I would not have made it to this point. I also would like to thank my thesis committee as well for their support and guidance during this endeavor. I thank the Watson Scholars Initiative for giving myself and my fellow Scholars the opportunity to attend AFIT and earn our Masters degree. I also appreciate the moral support that all my fellow Scholars and students gave me throughout the entire time we shared at AFIT. Finally, I am grateful for my parents and family without their constant encouragement, love, and support, I would not have the strength to endure the hard-times. v

7 Table of Contents Page Abstract... iv Acknowledgments...v Table of Contents... vi List of Tables... viii Table of Figures... ix 1 Introduction Background Problem Summary of Current Knowledge Literature Review Introduction Related Work Walker Satellite Constellations Beste Continuous Coverage Design Lang s LEO global coverage Doufer s zonal coverage optimization Emery et al s Constellations in matched inclination orbits Methodology Overview Parameters for an Eccentric Orbit Two-Body Orbit Approach Inclination Eccentricity Orbital Perturbation (J 2 ) Daylight Determination (4: 54-67) Matlab Algorithm Satellite to Target Range Results Single Satellite Coverage Inclination Effects Target Latitude of 10 Degrees Target Latitude of 33 Degrees Target Latitude of 50 Degrees Eccentricity Effects Varying Inclination and Eccentricity Satellite to Target Slant Range Inclination Comparison Eccentricity Comparison Coverage of Two Satellites Conclusion...61 vi

8 Page 5.1 Recommendations for Future work Bibliography...64 Vita...65 vii

9 List of Tables Page Table 1: Range of Orbital Elements Table 2: Slant range comparison between circular orbits at 33 and 41 degree inclination Table 3: Average Height and Average Slant Range Comparison of a satellite in a circular orbit with a semi-major axis of 350 km and a satellite in an elliptical orbit that has and eccentricity of e = with a semi-major axis of 575 km Table 4: A comparison of Daylight and Max Argument of Perigee for a satellite in a 350 km circular orbit at true anomaly of 0 and 180 degrees viii

10 Table of Figures Page Figure 1: Target to Satellite elevation geometry Figure 2: Satellite angle determination relative to the Sun s position to determine if pass is in daylight Figure 3: High-Level TACSAT Orbit Optimization Algorithm Flow Figure 4: Slant range depiction at the time the target is in view of the satellite Figure 5: Daylight Passes for a satellite at 100km altitude as inclination changes with respect to target latitude. Data collected over a 30-day period for each delta i. (Ω, ω are chosen to maximize daylight passes) Figure 6: Daylight Passes for a satellite at 350km altitude as inclination changes with respect to target latitude. Data collected over a 30-day period for each delta i. (Ω, ω are chosen to maximize daylight passes) Figure 7: Daylight Passes for a satellite at 600km altitude as inclination changes with respect to target latitude. Data collected over a 30-day period for each delta i. (Ω, ω are chosen to maximize daylight passes) Figure 8: Daylight Passes for a satellite at 800km altitude as inclination changes with respect to target latitude. Data collected over a 30-day period for each delta i. (Ω, ω are chosen to maximize daylight passes) Figure 9: The optimal inclination difference (vs. site latitude) increases with satellite altitude. This is due to the increase in the sensor footprint at the higher altitude Figure 10: Daylight Passes for a satellite at 100km altitude as inclination changes with respect to target latitude. Data collected over a 30-day period for each delta i. (Ω, ω are chosen to maximize daylight passes) Figure 11: Daylight Passes for a satellite at 350km altitude as inclination changes with respect to Figure 12: Daylight Passes for a satellite at 600km altitude as inclination changes with respect to target latitude. Data collected over a 30-day period for each delta i. (Ω, ω are chosen to maximize daylight passes) Figure 13: The optimal inclination difference (vs. site latitude) increases with satellite altitude Figure 14: The optimal inclination difference (vs. site latitude) increases with satellite altitude. This is due to the increase in the sensor footprint at the higher altitude. (Ω, ω are chosen to maximize daylight passes) Figure 15: The optimal inclination difference (vs. site latitude) increases with satellite altitude Figure 16: The optimal inclination difference (vs. site latitude) increases with satellite altitude Figure 17: The optimal inclination difference (vs. site latitude) increases with satellite altitude Figure 18: The optimal inclination difference (vs. site latitude) increases with satellite altitude ix

11 Page Figure 19: The optimal inclination difference (vs. site latitude) increases with satellite altitude Figure 20: Total number of daylight passes for different RAAN and argument of perigee combinations for a circular orbit with an inclination of 33 degrees and zero eccentricity Figure 21: Total number of daylight passes for different RAAN and argument of perigee combinations for a circular orbit with an inclination of 33 degrees and an eccentricity e = Figure 22: Daylight Passes comparison for a satellite with different eccentricities with equal inclination for a 30-day period Figure 23: Daylight Passes comparison for a satellite with different eccentricity and inclination for a 30-day perion Figure 24: Daylight Passes Shift for different RAAN values x

12 OPTIMAL COVERAGE OF THEATER TARGETS WITH SMALL SATELLITE CONSTELLATIONS 1 Introduction 1.1 Background The use of satellites has steadily increased since the start of the space age. Although there are many reasons for using satellites, the three main uses have been Earth observation (including weather), communication, and navigation. These uses require continuous coverage of the globe in order to gather and communicate information from around the world. These missions have become increasingly important for national defense, in which planners and commanders now desire global coverage in real time. The need for real time information for an up to the minute view of an area of conflict is vital for strategic purposes. When the military needs to go into an area of conflict, the first objective is to gain visibility of that area so as to observe enemy movements and plan operations. This military need can be met by sending reconnaissance aircraft to that area in order to get this information, but missions like this put aircraft and crew at risk. Satellites can gather similar reconnaissance information as an aircraft but much more safely because of the wide difference of altitude a satellite operates compared to an aircraft. These goals are hard to accomplish with the use of one satellite, so collections of satellites (known as a satellite constellation) became a primary focus of study. At high altitudes, such as geosynchronous orbits, continual coverage is possible with just a few satellites in a constellation, but as the satellite altitude is lowered (for example, to 1

13 improve imaging resolution), the number of satellites required for continual coverage rapidly increases. Therefore, it has been proposed that constellations could be designed for limited (theater) area coverage and optimized to maximize coverage over time. Currently all satellites in orbit are national assets, this means if a field commander in a theater needs to have information provided by a satellite, that commander needs to give this request to the organization who is in charge of operating that specific satellite. The information requested by the field commander needs to be real time imaging of the specific region of interest in order to have up to date information to carry out the essential mission. With the current process used for obtaining satellite imaging information, the imagery given to the field commander would not be an accurate real time image of the specific region of interest. This is due to the time spent requesting the information and waiting to receive the information needed; this time spent could be a critical component in planning and executing a successful mission. The TACSAT program in turn wants to look at the utility of one or more satellites that are directly controlled by the commander in a theater. 1.2 Problem A satellite able to perform the functions needed by the field commander has not yet been fielded for routine operations. The desired spacecraft need not be state-of-theart; but rather capable only of simple tasks of taking images of the theater of operations, and sending that information to the field commander in a timely manner. The cost of building a spacecraft for this specific mission should not be too high; it is the cost of launching the satellite that would be costly. Because of the high cost of launching a satellite into space it is important to find and use the optimal configuration for a satellite's 2

14 orbit. Using one satellite may supply adequate coverage of the region but using a small constellation of satellites, for example two satellites, can increase the coverage time. Finding the appropriate orbit for a single satellite, as well as for a two satellite constellation, is the primary purpose of this thesis. As a part of their study on the military utility of TACSATs, Emery et al. investigated using circular orbits to meet this criterion. The orbit optimization was only one aspect of the work; they also considered the logistical impact and the analysis of the use of satellite constellations for tactical needs. It was an AFRL sponsored thesis for a proof of concept development of a responsive, tactical, space based ISR system (4: 5). System Specifications for the TACSAT were provided; these specified different components of the TACSAT. The specification parameters provided dealt with the mission s life and duration as well as the orbital inclination used and area of interest for surveillance. The TACSAT system specification for the inclination was a matched to Theater latitude inclination. The theater in question is the Iraq Theater that has a latitude of 33 degrees. In addition, they were provided with the design parameters for the satellite. These included the orbit altitude and the eccentricity. The altitude of the orbit was set at 350 km and was to be circular; in other words, it had an eccentricity of zero. One of their goals was to use the specifications and parameters given to them to simulate the number of passes the satellite would have over the target. They needed to find an optimal orbit configuration for the satellite so as to provide optimal coverage of the target. A member of the group, Major David B. Smuck developed a Matlab program specifically for this purpose. Since one of the design parameters of the TACSAT was to be in a circular orbit, there were a number of orbital elements left as constants through 3

15 out the simulation. The elements left constant were the semi-major axis, the eccentricity, and the inclination. The right ascension of the ascending node (RAAN) and the argument of perigee were varied from 0 to 360 degrees. The RAAN and argument of perigee were varied in increments of 36 and 30 degrees, respectively. The program s main function was to quantify the total number of daylight passes over the target at different RAAN/argument of perigee combinations. The RAAN/argument of perigee combinations as well as the other orbital element values allowed Emery et al. to configure a set of orbits that would give the optimal coverage of the area of interest. The group first simulated for a single satellite. Once they acquired the optimal configuration for a single satellite, their next step was to find the optimal configuration for a constellation of satellites. Since their work only dealt with numerically simulating the coverage of a satellite constellation in a matched inclination circular orbit, the next step should be to see how non-matched inclination and elliptical orbits might affect the satellite constellation's optimal coverage of the target area. 1.3 Summary of Current Knowledge Several researchers have studied methods for designing satellite configurations for continuous global coverage. Work done by Beste dealt with the design of satellite constellations for optimal continuous coverage (1: 466). Lang looked at the optimization of low-earth orbit (LEO) constellations for continuous global coverage (5: 1199). J. G. Walker discussed different methods for a satellite constellation configuration that would give global coverage. These researchers discussed different techniques for configuring a satellite constellation that would give complete global coverage, but few have studied constellations with smaller coverage areas. One who did 4

16 was Doufer; he looked into the optimization of satellite constellations for zonal coverage (3:609). He used methods from Walker to produce his own configurations for coverage of specified latitude bands with a constellation of satellites. As discussed earlier, Emery et al. also investigated regional coverage, using a numerical approach to optimize small satellite constellations using circular orbits with inclinations matched to the target latitude. The purpose of this thesis is to expand upon the work of Emery et al. and quantify the coverage of a small satellite constellation by varying specific orbital elements. 5

17 2 Literature Review 2.1 Introduction The literature review discusses the different techniques used to construct LEO orbit satellite constellations that provide an optimal coverage of a region on the earth's surface. Satellites in geosynchronous or geostationary orbits (GEO) are not considered hear. Although GEO satellites provide excellent Earth coverage due to their high altitude and constant longitude, the 36,000 km altitude is too high for high resolution imaging missions. The review then discusses continuous coverage of areas the size of a few hundred kilometers in radius. Continuous coverage of the above mentioned areas is desired for the purpose of this thesis. Therefore, continuous global coverage will not be discussed in much depth. It will, instead, be referenced from time to time. There are many common orbit classes used by both military and commercial industries. They include GEO, medium Earth orbit (MEO), LEO, and highly elliptical orbits (HEO). GEO orbits have an orbital radius of over 42,000 km; these orbits provide good everyday coverage of one particular region of the earth. GEO orbits have an inclination angle near zero which means that the orbit s trajectory runs along the Equator. GEO satellites have an orbital period equivalent to one day, so that the region on Earth that the satellite s field of view covers remains the same. Typical missions for GEO satellites include communications and low resolution imaging. MEO satellites are typically either for scientific missions or navigation-related, like the global positioning system (GPS). HEO satellites are less common, but sometimes used for communications or low resolution imaging because they provide better coverage of near-polar latitudes than GEO satellites. LEO satellites provide less coverage because they have a lower 6

18 altitude than a GEO; their field of view over the earth s surface is smaller. However, lower altitude means better imaging resolution. Also LEO orbits generally have an inclination greater than zero degrees. The inclination is the angle the satellite s orbit path makes when it crosses the equator. Since a satellite on a LEO orbit can have an inclination greater than zero, the satellite makes a ground track on the Earth s surface that covers more latitude bands. In addition, a satellite in LEO orbit will have a greater angular speed compared to the Earth s rotation. This means that a satellite on a LEO orbit could pass over a certain region of the Earth many times in one day but it will not provide continuous coverage of that region. This is because once a satellite passes a certain region it may take multiple passes for the satellite to orbit over the same region again. Therefore, it is necessary to have more than one satellite to have a continuous coverage of a desired region of the Earth. This need for several satellites has led researchers to look into satellite constellations to provide what is desired. 2.2 Related Work Walker Satellite Constellations J. G. Walker has studied satellite constellations that would provide global coverage of the earth. He stated that circular orbits are preferred rather than elliptical orbits because circular orbits are more suitable for global coverage. He also discusses the importance of satellite constellation with multiple orbits by stating that "single orbit cannot provide either a regular polyhedral distribution or whole-earth coverage (7: 559). It is not only important to have multiple satellites in orbit but also to have them in different orbits to achieve the whole-earth coverage desired. He goes on to say that in choosing a satellite constellation, a designer needs to meet various constraints of the 7

19 system in order to be able to achieve the required standard of coverage (7: 559). He states that the best way to simulate a satellite constellation is to configure the constellation relative to a polyhedron distribution. Walker pointed out that even though it cannot be established in practice, "a satellite constellation in which the distribution of satellites on the spherical surface containing the [circular] satellite orbits corresponds to the vertices of a regular polyhedron (7: 560). What he meant is that the orbit of one satellite as well as it s relative distance from an adjacent satellite set up to relate to a polyhedron design. The use of this geometric configuration lets him establish three practical satellite constellations: delta patterns, sigma patterns, and omega patterns. Delta patterns contain a total number of satellites obtained by multiplying the total number of orbits being used by the number of evenly space satellites in each orbit (7: 563). He states that delta patterns have superior coverage characteristics. In regular satellite patterns the relative position of the satellites changes during an orbital revolution around the earth, but "the highly uniform nature of a delta pattern ensures that similar configurations recur frequently during one orbital period (7: 563). Another advantage of delta patterns is that the method of description is "independent of satellite altitude or orbital period (7: 563). Thus the pattern of the satellite constellation will remain unchanged regardless of the altitude or period (7: 563). The second practical satellite constellation follows the sigma patterns which is a subset of delta patterns with more attention to the path the satellite follows along the earth. Sigma patterns consist of patterns which "follow a single Earth-track which [does] not cross itself and [is] repetitive after [a certain amount of] days (7: 565) ". It is simply trying to simulate a sinusoidal pattern with that of the satellite ground track. Omega pattern on the other hand 8

20 takes into account that satellites will fail for some reason. Unlike the first two patterns which were uniform, omega patterns is used for non-uniform constellations. It is a subset of delta patterns by having a certain number of satellites that are needed for the configuration but actually using fewer satellites for the coverage needed. It will allow you to have extra satellites already in the satellite constellation at your disposal whenever there is a need to use them for whatever reason Beste Continuous Coverage Design Dr. David C. Beste discussed two satellite constellation designs that provide continual coverage of certain regions on earth. One satellite constellation design involved polar orbits; these are orbits that generally have a North to South trajectory with an inclination of 90 degrees. He stated that this polar orbit arrangement clusters the satellites in an optimal manner at the equator (1: 467) ". In other words the maximum coverage of polar orbit satellite constellations is centered on the equator. Dr. Beste derived his results for single coverage which meant he wanted to find the maximum coverage area of a particular region using the least amount of satellites and orbits. He then follows this first design with Full Coverage Beyond Latitude λ (1: 468). This second design was used to show the maximum regions these polar orbits covered between certain latitude, the north and south poles. These regions are beyond latitudes of positive and negative 30 degrees up to the north and south poles respectively. He later examined non-polar orbits, orbits that have an inclination of less than 90 degrees. From his results he concluded that the "polar-orbit configuration was superior (1: 469). Dr. Beste then derived a configuration that involved ideas of both polar and non-polar orbits by seeking a "three mutually orthogonal orbital planes (1: 469)". This configuration 9

21 consisted of four orbital planes and a total of 12 satellites; the results showed that the coverage area of this configuration was higher than a polar configuration of three orbits and 12 satellites (1: 469). He stated that in order to have continuous coverage of the earth there needs to be a considerable amount of overlap (1: 469). This means that the coverage areas of each satellite in each configuration need to have a section of their region also covered by an adjacent satellite Lang s LEO global coverage Dr. Thomas J. Lang discusses optimal LEO constellations for continuous global coverage (3: 1199). He states that even though a satellite constellation for this purpose in a LEO orbit would require a significant amount of satellites, it is a cost effective solution. Like the other researchers mentioned above, Dr. Lang suggested it was "important to optimize the constellation so as to find the minimum number of satellites required performing the mission (5: 1200)". He also stated that in many cases "non-polar constellations outperformed similarly sized polar constellations (5: 1200)". Because Dr. Lang focused on the continual global coverage of satellite constellations, his work reinforces the work by both Dr. Walker and Dr. Beste work Doufer s zonal coverage optimization Dr. F. Doufer deviates from the previous works by working on optimizing zonal coverage of satellite constellations instead of global coverage (3: 609). He states that many methods have been developed to determine the coverage of constellation patterns, but that all involved global coverage (3: 609). He says that constellations that are used for continual global coverage are not cost effective because of the uselessness of covering 10

22 the entire planet (3: 609). He categorizes the many methods for evaluating coverage into two categories, the semi-analytical methods and the numerical methods (3: 609). A semianalytical method is like Walker s satellite triplets or Rider s streets of coverage (3: 609). He states that the numerical method are normally more flexible and can usually deal with [evaluating more complex coverage objectives, which the previous method is inadequate of doing], but to the detriment of computation times (3: 609). For his work on zonal coverage, Doufer uses Walker s triplets method for global coverage analysis and extends it for evaluating and optimizing zonal coverage (3: 610). His objective is to precisely assess zonal coverage properties of any constellation pattern where all satellites are using circular orbits with identical orbital periods (3: 610). He first begins by discussing the coverage of a single satellite or in other words explains what the ground coverage of the satellite depends on. He states that the satellites ground coverage depends on the user and satellite altitudes, the minimal elevation angle and the distance from the user to the satellite (3: 611). He concludes his discussion of a single satellite coverage by saying that once the size of the satellite s ground coverage is known then it will be easy to deduce a satellite altitude form a specific minimal elevation angle [or vice versa] (3: 611). He then discusses global coverage and how Walker s satellite triplets technique alone cannot achieve a coverage assessment of a constellation pattern with non-global coverage (3: 612). He continues by discussing his zonal coverage method and states that it only deals with coverage objectives defined as latitude bands (3: 613). He also adds that for zonal coverage analysis, areas of interest are delimited by zonal boundaries and must comply with a specific coverage level (3: 613). This method expands on Walkers triplets method so that it can provide information when the center 11

23 point falls outside the target area even though the associated circumcircle overlaps the target area, he calls this the worst-seen point (3: 613). He provides examples for zones defined by latitude bands and states that longitude boundaries, while difficult to incorporate, are possible with his methods Emery et al s Constellations in matched inclination orbits The thesis research done by Emery et al dealt with regional coverage of smallsatellite constellations in with matched inclination LEO orbits (4: 1). Their work dealt with a specific region of interest where a satellite needed to take images of target areas within that region. They were given specific specifications that they used to develop a Matlab program that simulated orbits for satellites that would be used. This researched only dealt with satellite constellations that were on circular orbits and that had an inclination equal to the target region s latitude. The Matlab program they developed outputted orbital information for a single satellite in orbit for mission duration of one month. Within this month duration the program calculated the number of passes the satellite had over the targeted region for the thirty day duration. The given specifications for the imaging device on board the satellite constrained the total number of satellite passes to only the passes the satellite had over the target area in daylight. Their program had to deal with determining the satellite s position with respect to an Earth Centered Inertial frame which enabled them to determine the Sun s relative position every time the satellite passed over the target area. This let them know if the pass was during daylight or not. The orbit parameters that were used for the simulations were determine and used to provide the optimal amount of passes for a single satellite. The parameters that were left constant for the simulation were the semi-major axis, eccentricity, and inclination. 12

24 These constant parameters were use to output different combinations of right ascension of the ascending nodes and argument of perigee and provide the total number of daylight passes the satellite passed over that area. With the advancement of satellites and the higher cost of satellite launches, the optimum configuration of satellite constellations is necessary in order to be able to get the maximum use of the system. There have been many configurations that have been looked at, some concentrate on the orbits of the constellation, either being polar or nonpolar. Yet other researchers focus on the effect that the shape of the constellation can have on the amount of coverage it can achieve. There are many different criteria that need to be taken into account in order to choose the optimum configuration, but the most important is the amount of coverage that the satellite constellation can achieve. Most of the research that has been done during the years that deal with satellite constellations has dealt with continuous global coverage with a satellite constellation size of over twenty satellites. These papers don t necessarily consider optimizing coverage over one target area up until recently. 13

25 3 Methodology 3.1 Overview Satellite constellations can be used to obtain imagery data from a region of interest. As mentioned earlier, the motivation behind this research was to determine the optimal configuration of a satellite constellation to maximize the number of daylight passes over a target area. The initial analysis was done using a single satellite optimizing the coverage for the target area. Once coverage of a single satellite was optimized, then the implementation of two or more satellites was considered. The optimal configuration for a single satellite was determined with a numerical search method varying inclination and eccentricity for the satellite s orbit. Since the satellite uses an optical imaging device, only daylight passes were of interest. Varying the orbital parameters yielded different values for satellite daylight passes over the target area. These passes were collected for a period of 30 days. The distance between the satellite and the target, slant range, was also a parameter with which the optimal configuration was determined. The configuration that had the maximum number of daylight passes while remaining within an acceptable slant range was deemed the optimal orbit. After obtaining the optimal orbit configuration for a single satellite the same process was done for two or more satellites in the constellation. 3.2 Parameters for an Eccentric Orbit After using the inclination to affect the total number of daylight passes, the next step was to observe the effects of a non-zero eccentricity had on the total number of passes. 14

26 For non-zero eccentricity simulations, the minimum and maximum altitudes of the satellite needed to be known. The satellite specifications stated that the imaging device on board the satellite provides the greatest resolution at an altitude of 350 km, but it still provides acceptable resolution up to an altitude of 800 km. From these specifications, the minimum and maximum altitudes of the satellite were set. The perigee is defined as the minimum altitude of the satellite plus the radius of the Earth, 6378 km. On the other hand, the apogee is defined as the maximum altitude of the satellite plus the radius of the Earth. Since the minimum altitude was 350 km, the perigee of the satellite was 6728 km. The maximum altitude that the satellite can be at was 800 km, so the apogee was 7178 km. R = 6728 km p R = 7178 km a where R p and R a is the radius of the perigee and apogee, respectively. After perigee and apogee of the orbit were determined, the next step was to find the eccentricity. Using the following equations for the perigee and apogee of an orbit, R = a(1 e) (1) p R = a(1 + e) (2) a the eccentricity e was determined. Equation (1) was modified to solve for the semi-major axis in the following equation. R p a = (1 e) (3) Taking Equation (3) and substituting it into Equation (2) yielded, 15

27 R a Rp = (1 + e) (1 e) (4) Equation (4) was modified to solve for the eccentricity and the following equation was obtained. ( Ra Rp) e = ( R + R ) a p (5) Using the values of 6728 km and 7178 km for R p and R a respectively, Equation (5) generated the eccentricity value of e = Once the eccentricity was obtained, the next step was to determine the corresponding semi-major axis. Using Equation (1) along with the values for the perigee and the eccentricity, the value for the semi-major axis was determined to be a = km. The perigee, apogee, semi-major axis, and the corresponding eccentricity were needed to determine if configuration was acceptable. The value for eccentricity was used in the algorithm to determine if it produced a greater total number of daylight passes than a zero eccentricity (circular orbit) configuration. 3.3 Two-Body Orbit The orbital motion of a spacecraft around the Earth is frequently described by the dynamic associated with what is known as the two-body problem. The name stems from the assumption that the motion can be modeled as two point masses orbiting under their mutual gravitational attraction (8: 45). The two body problem helps determine the position and velocity of an object in orbit. Any particular orbit is completely determined by six orbit elements: semi-major axis, eccentricity, inclination, right ascension of the 16

28 ascending node (RAAN), the argument of perigee, and mean anomaly. Under the assumptions of two-body motion, five orbital elements stay constant as shown below, at () = at ( o ) (6) et () = et ( o ) (7) it () = it ( o ) (8) Ω () t =Ω ( t o ) (9) ω() t = ω( t o ) (10) The semi-major axis, a, determines the orbit s size while the eccentricity, e, determines the orbit s shape (8: 57). The inclination, i, is the angle that the orbit makes with respect to the Earth s equator. The right ascension of the ascending node, Ω, and the argument of perigee, ω, completes the definition of the orbit s orientation with respect to inertial space. The mean anomaly changes due to the mean motion because the mean anomaly gives the position of the satellite within the orbit (6: 1). The mean anomaly, on the other hand does change. Equation (11) shows the relationship between the mean anomaly and time M () t = M + n( t ) (11) where M is the mean anomaly, M o is the mean anomaly at epoch, and n is the mean motion or the orbit s angular frequency (4: 56). o o 3.4 Approach A numerical approach was used to ascertain the total number of passes a satellite made over a specific target. In order to determine the optimum orbital configuration, the Matlab program developed by Emery et al. was modified to account for the variance in 17

29 inclination and eccentricity. Emery et al. concluded that improved data transmission between the satellite and receiving ground station occurred when the receiving station was placed at a latitude lower than the satellite s inclination (4: 230 ). From their conclusions, an inclination that is higher than the target latitude would yield an increased number of daylight passes. The eccentricity was also varied to determine if similar effects occurred. When a satellite constellation is in an elliptical orbit, the altitude of the satellites with respect to the target location will vary; this is different from a circular orbit where the altitude remains relatively constant. The minimum and maximum altitudes are designated as the perigee and apogee, respectively. The eccentricity of the orbit is defined by the perigee and apogee. Eccentricity for elliptical orbits range between zero and one, zero being circular and one being parabolic. When the eccentricity is greater than zero, the satellite s orbit will be affected so as to pass over the target area at different elevations. Passes will occur at different altitudes ranging between the perigee and apogee. It is important to know the orbit s apogee because the resolution of the images is dependent on the satellite s altitude. Lower altitude passes provide better image resolution than higher altitudes due to reduced distances between the satellite and target. 3.5 Inclination The code developed by Emery et al. used an inclination of 33 degrees. This inclination was chosen to match the target latitude, and is often referred to as a matched inclination orbit. The inclination was varied starting at the target latitude of 33 degrees and initial altitude of 100 km. This inclination variance ( i) was defined as the difference between inclination and target latitude. Next the same inclination variation process was completed for different altitudes: 350, 600, and 800 km. The i that yielded 18

30 the maximum number of daylight passes was plotted against its respective altitude. This was conducted to determine if an increasing trend for i versus satellite altitude occurred, which was important because the total daylight passes is expected to increase with increasing i and altitude. This entire procedure, from varying i at different altitudes to plotting the i with maximum number of daylight passes at their respective altitudes, was repeated for target latitudes of 10 and 50 degrees. 3.6 Eccentricity Since a non-zero eccentricity was used, the satellite orbit was elliptical. The semi-major axis is defined as the average of the perigee and apogee. Only one eccentricity value was focused on in this research. Because of the specifications and design parameters of the satellite and imaging device, the value of the eccentricity used was constrained. The maximum altitude used for the satellite was 800 km. This altitude was chosen because it is the maximum distance at which the imaging device payload can provide acceptable imagery data. The minimum altitude was set to be 350 km for the satellite s orbit. This value was specified as the minimum orbit altitude by the sponsors for previous studies. From these minimum and maximum altitudes, the eccentricity value used was e = The reason why this eccentricity value was used is because of the corresponding semimajor axis, perigee, and apogee values. For an eccentricity value of e = 0.032, the perigee and apogee are 6728 km and 7178 km respectively. Both the argument of perigee and the RAAN play an important part on a satellite's orbit. The argument of perigee "defines where the low point, perigee, of the orbit is with respect to the Earth's surface" (6: 1). The RAAN "defines the location of the ascending 19

31 and descending orbit locations with respect to the Earth's equatorial plane (6: 1). Both of these orbital elements are involved in determining the position of the satellite with respect to the Earth. 3.7 Orbital Perturbation (J 2 ) There are many forces acting on a satellite in addition to the point mass gravity force described above. These include air drag, third body effects, and gravitational effects caused by a non-spherical Earth. Each of these perturbing forces acts to slightly change the two body solution. In this study, only the greatest of these effects was considered. This effect is due to the non-spherical, ellipsoid nature of the Earth; this is often termed the J 2 effect (9: 20). This affects all the equations discussed above; J 2 affects the semi-major axis, the eccentricity, and the inclination in similar ways. This effect produces a periodic change to these three orbital parameters. Even though they are changing, the average value for each parameter over a period of thirty days is nearly constant, as if it were a simple two body orbit. On the other hand, the mean anomaly, RAAN, and the argument of perigee all experience the J 2 effect in different ways. When J 2 is taken into consideration, the equations for these three parameters must be modified. In a regular two-body problem the mean anomaly was the only orbital element that varied with time, but if the J 2 effect is taken into account, the equation for the mean anomaly changes as follows M () t = M( to) + M o t+ nt (12) Here, the mean anomaly depends on the position of the satellite at time equals to zero as well as the mean motion at any given time. The RAAN and the argument of perigee are 20

32 similarly affected when the J 2 perturbation is taken into consideration. Because of the perturbation, Equations (9) and (10) are changed as follows Ω () t =Ω ( t ) +Ω t (13) o ω() t = ω( t ) + ω t (14) They each depend on the initial value for the RAAN, Ω ( t o ), and argument of perigee, ω ( t o ), but an additional value needs to be added for the J 2 perturbation effect, which are o Ωt and ω t. Even though the J 2 perturbation effects on the orbital elements are minuscule, they were taken into consideration because of the mission length. If a satellite s orbit needed to be calculated for a period of one or two days, the J 2 perturbation effect could be neglected. This J 2 value is small enough that using the orbital equations for a two body problem would be sufficient. However, for the 30-day period investigated in this study, the effects were non-negligible. In this case, neglecting the J 2 perturbation effect on the mean anomaly, RAAN, and the argument of perigee would produce inaccurate results. The original Matlab code, written by Smuck, was based on similar equations in which M,, 0 Ω and ω were empirically determined by comparing results to similar predictions by Satellite Tool Kit. These constants have since been compared to the theoretical values, and provided reasonable approximations. The number of daylight passes over a target, for a given orbit, was determined using a Matlab algorithm as part of a previous study that examined the problem with an orbit inclination matched to the target s latitude. This algorithm identifies pseudooptimal initial conditions for a single satellite that maximized the number of daylight 21

33 passes for a given time span (4: 51). This algorithm took into consideration the six orbital elements. Inclination, eccentricity, and semi-major axis, were set as constants. RAAN and argument of perigee were varied from zero degrees to 360 degrees with a step size of 36 degrees and 30 degrees respectively. The mean anomaly was set to zero degrees. The RAAN and argument of perigee combinations yielded a total number of daylight passes over the specified target at each RAAN step (4: 52). Since the satellite was required to have visibility of the target during daylight, in order for the pass to be counted, the algorithm required the capability to distinguish such passes. 3.8 Daylight Determination (4: 54-67) In order to determine if the satellite s pass over the target site was during daylight, the positions of the target site, the satellite, and the position of the sun, with respect to an Earth Centered Inertial (ECI) reference frame, were tracked throughout the simulation. The code started by determining the position of the target vector. From the latitude, longitude, and altitude of the target, the target s position vector in an Earth Centered Fixed (ECF) frame was determined. This was done by using the following equations 22

34 ( N + h)cos( φ)cos( λ) RSiteECF = ( N + h)cos( φ)sin( λ) 2 [ N(1 a ) + h]sin( φ) (15) where N = α R φ 2 2 (1 sin ( )) (16) 2 2 α = 2 f f (17) f = Earth s flattening factor R = Earth s radius φ = geodetic latitude λ = east longitude h = geodetic altitude After determining the position of the target, the next step was to compare it to the position of the TACSAT. The satellite s position was calculated with respect to an ECI coordinate system at a given simulation epoch by transforming the classical orbital elements equations, which were discussed in the previous section, at any point in time. The satellite s ECI position vector was calculated for a period of 30 days at one minute increments. The next step was to convert all the position vectors calculated to a single coordinate frame; this was done to evaluate the interaction of the system. In order to do this, the epoch time was determined in Julian Centuries using a reference Julian date of 1 January 2000 at 1200Z. Next, the Greenwich Mean Sidereal Time (GMST) at the simulated epoch was calculated by using the Earth s rotation rate and the time past epoch 23

35 at each time step. The resulting GMST was then transferred from ECF to ECI coordinates using the following equation. cosgmst singmst 0 M ECF ECI = singmst cosgmst (18) Next, the target to satellite elevation was calculated in order to determine if the satellite is within view of the [target]. Figure 1: Target to Satellite elevation geometry Figure 1 shows how the satellite s elevation with respect to the target was calculated. el π R ir = siteecf sitetosat arccos 2 RsiteECF R sitetosat (19) 24

36 Using Equation (19), the elevation angle was determined; if a positive elevation angle is calculated, then the satellite is in view of the target. Once the target and satellite positions were known, the next task was to calculate the Sun s position vector so as to determine if the satellite s pass over the target was in daylight. The time in Julian centuries referencing 1 January 2000 was used for this calculation. Emery et al. began by calculating any time past the simulation epoch T t / = T0 + (20) where T = time (Julian Centuries) T 0 = epoch time (Julian Centuries) and the same t = time (sec) that was used to determine the satellite s position vector. Next, Emery et al. calculated the mean longitude and the mean anomaly of the Sun, as m well as the distance from the Earth to the Sun ( λ, M and the following: longitude of the ecliptic r respectively) to acquire m λ = λ sin( M ) sin(2 M ) (21) obliquity of the ecliptic ε = T (22) Once both the longitude and the obliquity of the ecliptic were known, the ECI position vector of the Sun was calculated using R = r cos( λ ) cos( ε)sin( λ ) sin( ε)sin( λ ) (23) 25

37 Once the ECI position vector of the Sun was known, then the angle between the Sun position vector and the target position vector could be calculated. If this angle was less than 90 degrees, the target would be in daylight at the time of the pass. Figure 2: Satellite angle determination relative to the Sun s position to determine if pass is in daylight. Figure 2 shows how the daylight determination of the satellite over the target was calculated. The value for θ was the determining factor whether the satellite s pass over the target was in daylight or not. The value for θ was calculated by using the following equation: cosθ = R ir target i target R r (24) The angle θ depends on the position vector, R, of the Sun and the position vector of the target, r target. The angle θ needed to be less than 90 degrees in order for the pass to occur during daylight. 26

38 3.9 Matlab Algorithm A block diagram of the algorithm adapted from the Emery et al. thesis is shown below in Figure 3 (4: 58). Define initial Calculate TACSAT Is Target visible to Yes Is Target in Daylight Yes Was Target visible and in daylight at previous time step Yes No No No Increment pass Increment to next time Figure 3: High-Level TACSAT Orbit Optimization Algorithm Flow The first step in the algorithm was to define the initial conditions for the orbital elements. Next, the position of the satellite was calculated. The algorithm then determined if the target was visible to the satellite. If so, it proceeded to the next step; if not, the algorithm incremented to the next time step and calculated the new satellite position until the target was view of the satellite. Once the target was visible to the satellite, the algorithm checked if the target was in daylight. If true, it proceeded to the next step; if false, then it incremented to the next time step and calculated the new position of the satellite. If the target was visible during daylight, a pass counter was incremented and the process repeated, by then calculating the next satellite position. This process was repeated for each combination of RAAN and argument of perigee, summing all daylight passes, and 27

39 finally, displaying the results on a graph. The output displayed the daylight pass totals versus the corresponding RAAN value. This algorithm was used by Emery et al. to determine maximum daylight passes with a matched inclination to the target latitude for circular orbits. However, in this study, it was modified and used to determine if varying the inclination and eccentricity had a positive effect on the total number of daylight passes. The inclination was varied in one degree increments, starting from the initial match inclination. As mentioned in a previous section, it was hypothesized that if the inclination of the orbit was greater than the target latitude, the total number of daylight passes would also increase when compared to the number of daylight passes for a matched inclination orbit. The algorithm was iterated for each inclination value. If each greater inclination value yielded more daylight passes, the process continued to be implemented until the total number of passes peaked Satellite to Target Range The satellite did not always take pictures of the target when it was passing directly above; most of the time the images were taken when the satellite was closest to the target for the given pass. Since some of the tests that were conducted involved varying the inclination and the eccentricity, the satellite s passes were sometimes further away from the target. Therefore, the distance between the satellite and the target needed to be known, especially in an eccentric orbit, where the satellite has a minimum and a maximum altitude. The distance between the satellite and the target was called the slant range. This distance was calculated whenever the target was in view of the satellite. 28